Research on Dynamical Systems, Chaos and Fractals Modeled by Billiard Systems in Aspect of Global Analysis

本文改进了无食二维离散台球Lipschitz连续不变叶理存在性的证明，使之更有建设性，更初等。这可能会在应用方面取得新的进展。我们引入了Busemann测地线的几何，研究了台球在其位形空间中的轨迹。特别地，我们发现了平行线和周期轨线之间的一些关系。此外，我们观察到台球系统在一个封闭曲线中的有趣现象，|X/a| ^r+| y/B| ^r=1（r>1），计算机模拟。它的Poincare映射提供了大量的信息，可以证明这些现象.在被视为经典周期台球玩具模型的双曲和间歇映射中，发现输运可以用Frobenius-Perron算子的谱来表征.进一步研究了边界元方法与量子台球中散射问题的关系，研究了复动力学中不同周期点的重整化。我们能够定义一个新的映射空间，该空间在抛物重整化及其扰动下是不变的。我们观察到可数到一个分段可逆马尔可夫系统的大偏差。特别地，在一定条件下，我们给出了（水平2）大偏差上界和指数递减性质。此外，我们还研究了大偏差律的多重分形形式，并成功地证明了波义耳和Maass的猜想。我们可以构造一个单参数复杂结构族，其临界点位于递归和瞬态切换到另一个点处。我们表明，自由变形，以保持复发点是相对较低的。作为与量子混沌等相关的课题，我们一直在研究算子代数本身的理论，并得到了一些结果。

We have improved the proof of the existence of Lipschitz continuous invariant foliations for two dimensional dispersing billiards without eclipse and make it more constructive and more elementary. This may give a new progress in applications. We introduced the geometry of geodesies due to Busemann to study the billiard ball trajectories in their configuration spaces. In particular, we found out some relations between parallels and periodic trajectories. Moreover, we observed interesting phenomena of billiard systems in a closed curve given by |x/a|^r+|y/b|^r=1(r>1) with computer simulations. Its Poincare map gives a lot of information, by which those phenomena may be proved.In hyperbolic and intermittent maps regarded as toy models of classical periodic billiards, transports are found to be characterized by the spectra of the Frobenius-Perron operator. Furthermore, we researched the relationship between boundary elements method and scattering problems in quantum billiards.We studied the renormalization associated to indifferent periodic points of complex dynamics. We were able to define a new space of maps which is invariant under parabolic renormalization and its perturbations.We observed large deviations for countable to one piecewise invertible Markov systems. In particular, we showed the(level 2)upper large deviation bounds and exponential decreasing property under certain conditions. Moreover, we researched multifractal version of large deviation laws.Further, we have succeeded to prove the conjecture by Boyle and Maass. We can construct a one-parameter family of complex structures with critical point at the point the recurrence and the transience switch to the other. We showed that freedom of deformation to preserve recurrence at the point is relatively low. As related topics to Quantum Chaos and so on, we have been studying the theory of operator algebras itself and obtained several results in the subject.